Variational approach on whole body SPECT/CT registration and zipping

ABSTRACT

A method and a system for implementing the method for simultaneously registering and zipping a multiple scan whole body SPECT/CT image. The method includes the steps of simultaneously registering and zipping multiple input images and re-sampling the registered images. The step of simultaneously registering and zipping multiple input images is accomplished by initially aligning the images to be registered with each other, aligning the images with a reference image, and adjusting the alignment of the images with each other.

BACKGROUND DESCRIPTION

1. Technical Field

This invention relates generally to SPECT and CT imaging. Specifically,it relates to registering SPECT images and CT images of the same patientregions, and “zipping” together SPECT images of different portions of awhole body scan to provide a single whole body image.

2. Background of the Invention

When taking whole body Single Photon Emission Computed Tomography(“SPECT”) and Computed Topography (“CT”) scans, in many machines thedetector's field of view (“FOV”) is limited. It is therefore oftennecessary to take several separate scans for SPECT, which overlap in thez direction (see FIG. 1), at two or more different positions withrespect to a patient. These separate scans must then be “zipped” orappended together after reconstruction.

When such zipping takes place, it is usually impossible to determine theproper zipping position in the overlapping region where zipping occurs,in the z direction (see FIG. 1). When two adjacent images are sooverlapped, the proper dividing line could be anywhere in theoverlapping region. Presently there are no satisfactory methods todetermine this position and to zip the two images together based eitheron relative bed positions or on image positions.

Even in cases where the zipping position can be approximated, i.e. whenthe full reconstruction range is used in the overlapping region, otherfactors may hinder a satisfactory zipped whole body image. These factorsmay include: bed deflection, patient motion, and image edge handling in3D reconstruction algorithms with CT attenuation correction.

In many current methods, image registration (i.e., between the SPECT andCT images) and zipping (i.e., of two SPECT images of overlappingadjacent patient regions) are done completely separately. Auto-zippingis done after the multiple whole body SPECT images have beenindividually registered with the CT image. The separate registration andzipping processes do not generate satisfactory whole body images.

To solve these problems, it is desired to merge the registration taskand the zipping task into a single optimization task.

SUMMARY OF THE INVENTION

Therefore, according to the present invention a method forsimultaneously registering and zipping a multiple scan whole bodySPECT/CT image is provided. The method includes the steps of (a)simultaneously registering and zipping multiple input images and (b)re-sampling the registered images. The step of simultaneouslyregistering and zipping multiple input images is accomplished by (i)initially aligning the images to be registered with each other, (ii)aligning the images with a reference CT image, and (iii) adjusting thealignment of the images with each other.

In order to determine the best registration, which is used to generate aregistered output, the method uses the equation:

M _(total)(φ):=Σ_(j) M(U,V _(j)∘φ_(j))=max,

which is subject to

V _(total)(φ):=Σ_(i≠j) v(V _(i)∘φ_(i) ,V _(j)∘φ_(j))=min

where U is a CT reference image, V is a set of SPECT images, and φ is atransform.

Further provided is a system for implementing the method that includes aSPECT/CT scanning device, a processor that receives scans from theSPECT/CT scanning device, and software that inputs multiple images,simultaneously registers and zips the images and outputs a single,unified, registered image.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention will now be described in greater detail in the followingby way of example only and with reference to the attached drawings, inwhich:

FIG. 1 is an example of two SPECT scans with an overlapping region.

FIG. 2 is an example of unsatisfactory whole body zipping for two SPECTscans.

FIG. 3 is a diagram of the method for simultaneously registering andzipping a multiple scan whole body SPECT/CT image

FIG. 4 is a diagram of an algorithm for executing the method.

FIG. 5 is a system for simultaneously registering and zipping a multiplescan whole body SPECT/CT image.

DETAILED DESCRIPTION OF THE INVENTION

As required, disclosures herein provide detailed embodiments of thepresent invention; however, the disclosed embodiments are merelyexemplary of the invention that may be embodied in various andalternative forms. Therefore, there is no intent that specificstructural and functional details should be limiting, but rather theintention is that they provide a basis for the claims and as arepresentative basis for teaching one skilled in the art to variouslyemploy the present invention.

FIG. 1 depicts two whole body SPECT scans 110 and 120 of overlappingadjacent regions of a patient, where the solid lines represent the imagereconstruction range for each image. It is desirable to zip these twoSPECT scans 110 and 120 together to create one image. The two SPECTscans 110 and 120 will be zipped together somewhere in the overlappingregion 140. However, the proper dividing line 130 could be anywhere inthe overlapping region 140. If the wrong dividing line 130 is chosen, anunsatisfactory final image will be produced (see FIG. 2).

FIG. 3 diagrams the method 300 according to one embodiment of thepresent invention, which may find the proper dividing line 130 fromwhich to zip the SPECT images together to form a single whole bodyimage. The two main processes in the method 300 are (1) simultaneousregistration and zipping (340) and (2) re-sampling (350). Thesimultaneous registration and zipping 340 may include imputing areference CT image 310 and imputing whole body SPECT scans 1 (320)through K (330). The simultaneous registration and zipping 340 may notonly determine the best alignment (registration) between each individualSPECT image 320 through 330 and the reference CT image 310, but also maydetermine the best alignment among the K SPECT images 320 through 330themselves. After the simultaneous registration and zipping 340 iscomplete, the registered multiple images 360 may be re-sampled inprocess 350. The re-sampling 350 may sample the multiple registeredimages 360 (which may have overlaps) to generate a single unified output370.

FIG. 4 depicts an algorithm 400 for completing the method 300. Thealgorithm may consist of the steps of (1) scanning the SPECT images(“V₁, . . . , V_(K)”) 410, (2) finding an initial alignment among V₁, .. . , V_(K) images 420, (3) imputing a reference CT image (“U”) 480, (4)maximizing the total distance between U 480 and each SPECT image(“V_(i)”) 430, (5) minimizing the overall variation among overlaps ofV₁, . . . , V_(K) 450 (6) repeating (440) the maximizing 430 andminimizing 450 steps until the best registration is found, (6)re-sampling 460, and (7) registering an output 470.

Finding the initial alignment of the K images V₁, . . . , V_(K) 420 maybe done based on either bed positions or image positions of the Kimages. The maximizing 430 and minimizing 450 steps may be formulated asan optimization problem as follows.

Let R³ denote the usual three-dimensional Euclidian space. An image maybe defined as a function from R³ to R which satisfies certain regularityconditions. Given two images U and V, where U is a reference image and Vis the image to be registered towards U, the objective of theregistration between these two images is to find a proper transformation

φ:R³→R³

such that U and V∘φ are best matched in accordance with a certainobjective measure, where V∘φ denotes the registered version of V withV∘φ(x)=V(φ(x)) for x∈R³.

In a multiple input registration and zipping setting, there may be onereference image U (the CT image) and a set of K SPECT images {V_(j)} tobe registered (the multiple whole body SPECT images). It may benecessary to find K best transformations {φ_(j)} under certainoptimization criteria, where each φ_(j) represents the best registrationbetween U and V_(j). The set of functions {φ_(j)} cannot be foundseparately because their domains have overlaps in general, and those arethe regions where transformations need to be adjusted to make the bestzipping for the neighboring two images.

If φ=(φ₁, . . . ,φ_(K)), the maximizing (430) and minimizing (450) stepsmay be formulated as an optimization problem as follows:

Given one reference image U:R³→R³ and a set of K images {V_(j)} to beregistered, where V_(j): R³→R³, j=1, . . . , K, find a transformation φsuch that

M _(total)(φ):=Σ_(j) M(U,V _(j)∘φ_(j))=max

subject to

V _(total)(φ):=Σ_(i≠j) v(V _(i)∘φ_(i) ,V _(j)∘φ_(j))=min   (1)

where

M(U,V_(j)∘φ_(j)) measures the similarity between the reference image Uand the transformed image V_(j)∘φ_(j); M_(total) is the sum of allM(U,V_(j)∘φ_(j)); v(V_(i)∘φ_(i),V_(j)∘φ_(j)) measures the variationbetween the two registered images V_(i)∘φ_(i) and V_(j)∘φ_(j) at theiroverlapped region; and V_(total) is the sum of allv(V_(i)∘φ_(i),V_(j)∘φ_(j)).

One implementation for the optimization of problem (1) is to set theobjective functional as

J(φ):=−M _(total)(φ)+λV _(total)(φ)   (2)

and search for φ* such that J(φ*)=min, where λ is a constant to bedetermined. A gradient based steepest descent method may be used to seekthe minimum of the functional. First the gradient ∇J(φ) may becalculated, and then updates in the search for the optimaltransformation φ may be made according to

φ_(n+1)=φ_(n) −μ∇J(φ_(n)),μ>0,n=1,2,3, . . .

where μ is a constant used to control the convergence rate.

Thus, this registration algorithm searches for the best registration φbetween the set of images {V_(j)} and the reference image U in such away that the individual images V_(j)(1≦j≦K) are optimally aligned withrespect to the reference image U (in the sense of M_(total)=min).

Once the best registration φ has been found, it may be used in the finalre-sampling operation to generate a registered output. Note that in theconventional image registration setting where the re-sample is based onone transformation function φ only, the multiple input re-samplealgorithm in this operation must handle the multiple transformationfunctions {φ_(j)}. In particular, interpolation is needed in theoverlapped domain of the functions.

Often some type of regularization is needed because the imageregistration problem is ill-posed.

Let the transformation function φ:R³→R³ be the deformation map definedby

φ(x)=x+u(x)

where u is a proper function from R³ to R³.

For the similarity measure M between two images U and V, one may use thepopular mutual information defined by

${M\left( {U,V} \right)} = {\int_{R^{3} \times R^{3}}^{\;}{{p_{U,V}\left( {u,v} \right)}\log \frac{{p_{U}(u)}{p_{V}(v)}}{p_{U,V}\left( {u,v} \right)}\ {\left( {u,v} \right)}}}$

where p_(U) and p_(V) are the probability densities of the pixel valuesof the images U and V, respectively; p_(U,V) is the joint probabilitydensity of the pixel values of images U and V.

For the variation measure v between two overlapped images F and G, onecan use the sum of the squared difference defined by

v(F,G)=∫_(O)(F(x)−G(x))² dx

where O denotes the overlapped region between the two images.

Under these notation, equation (1) can be formulated as the followingvariational problem:

minimize J(u ₁ , . . . ,u _(k))+rS(u ₁ , . . . , u _(k))   (3)

where J is defined as in equation (2), S is a regularization term, andr>0 is a regularization parameter. In many cases, the regularizationterm S can be defined as a bi-linear form of B:

S(u ₁ , . . . ,u _(k))=Σ_(j=1) ^(K)∫_(D) <B(u _(j)),B(u _(j))>dx,D⊂R ³

where B is a differential operator, and <,> denotes the inner product inL₂(R³). L₂(R³) is the completion of the continuous functions withrespect to the L₂-norm. For example, for elastic registration, theintegral term in the above expression can be represented as

${\int_{D}^{\;}{{\langle{{B\left( u_{j} \right)},{B\left( u_{j} \right)}}\rangle}\ {x}}} = {\int_{D}^{\;}{\left\{ {{\frac{\alpha}{4}{\sum\limits_{i,{j = 1}}^{3}\left( {{\partial_{x_{i}}u_{j}} + {\partial_{x_{j}}u_{i}}} \right)^{2}}} + {\frac{\beta}{2}\left( {\nabla{\cdot u}} \right)^{2}}} \right\} \ {x}}}$

where α and β are the so-called Lamé constants, and ∇ is the divergenceoperator. Note that u is the function used to define the non-rigidtransformation function φ(x)=x+u(x). In a multiple input registrationsetting, with K input images to be registered, there should be K suchu's.

Using a proper discretization technique, the regularized minimizationproblem can be implemented as an iterative algorithm.

FIG. 5 shows an example system 500 that uses the method 300. The systemmay be comprised of a SPECT/CT scanning device 510 and a processingdevice. The processing device may obtain scanned images from thescanning device 510 and may run software that implements the algorithm400 to output a single registered and zipped whole body image. Thesystem may include a monitor 520 for displaying data, operatinginstructions, etc. from the processing device.

1. A method for simultaneously registering and zipping a multiple scanwhole body SPECT image with a CT image of a same region, comprising: (a)initially aligning multiple input SPECT images with each other; (b)aligning each of said multiple input SPECT images with said CT image;(c) adjusting alignment between said SPECT images based on alignmentwith said CT image; and (d) re-sampling the registered images to obtaina single output zipped image.
 2. The method of claim 1, wherein step (a)includes: (i) initially aligning the images to be registered with eachother; (ii) aligning the images with a reference image; and (iii)adjusting the alignment of the images with each other.
 3. The method ofclaim 2, wherein step (i) is based on bed positions.
 4. The method ofclaim 2, wherein step (i) is based on image positions.
 5. The method ofclaim 2, wherein the reference image is a CT.
 6. The method of claim 2,wherein steps (ii) and (iii) are repeated until a final alignment hasbeen attained.
 7. The method of claim 6, wherein steps (ii) and (iii)are accomplished by finding a transformation φ wherein:M _(total)(φ):=Σ_(j) M(U,V _(j)∘φ_(j))=max, subject toV _(total)(φ):=Σ_(i≠j) v(V _(i)∘φ_(i) ,V _(j)∘φ_(j))=min wherein:M(U,V_(j)∘φ_(j)) measures the similarity between the reference image Uand the transformed image V_(j)∘φ_(j); M_(total) is the sum of allM(U,V_(j)∘φ_(j)); v(V_(i)∘φ_(i),V_(j)∘φ_(j)) measures the variationbetween the two registered images V_(i)∘φ_(i) and V_(j)∘φ_(j) at theiroverlapped region; and V_(total) is the sum of allv(V_(i)∘φ_(i),V_(j)∘φ_(j)).
 8. The method of claim 7, wherein thetransformation φ is found by searching for a φ* such that J(φ*)=minusing the equation:J(φ):=−M _(total)(φ)+λV _(total)(φ) wherein: λ is a constant to bedetermined.
 9. The method of claim 8, wherein φ* is found by calculatingthe gradient ∇J(φ) and updating the search for the optimaltransformation φ according to:φ_(n+1)=φ_(n) −μ∇J(φ_(n)),μ>0,n=1,2,3, . . . wherein: μ is a constantused to control the convergence rate.
 10. The method of claim 7, whereinthe transformation φ (for φ(x)=x+u(x)) is found by minimizing:J(u ₁ , . . . ,u _(k))+rS(u ₁ , . . . ,u _(k))wherein:J(φ):=−M _(total)(φ)+λV _(total)(φ); wherein, λ is a constant to bedetermined; S is a regularization term; and r>0 is a regularizationparameter.
 11. The method of claim 10, wherein:S(u ₁ , . . . ,u _(k))=Σ_(j=1) ^(K)∫_(D) <B(u _(j)),B(u _(j))>dx,D⊂R ³wherein: B is a differential operator; and <,> denotes the inner productin L₂(R³); wherein L₂(R³) is the completion of the continuous functionswith respect to the L₂-norm.
 12. The method of claim 1, furthercomprising: (c) generating a registered output.
 13. A system forsimultaneously registering and zipping multiple scan whole body SPECT/CTimages, comprising: a SPECT/CT scanning device; a processor incommunication with the SPECT/CT scanning device; and software executingon the processor, wherein the software inputs multiple SPECT images anda CT image of the same region, simultaneously registers and zips themultiple SPECT images and registers the CT image with the SPECT images,and outputs a single, unified, registered image.
 14. The system of claim13, wherein the software: initially aligns the images to be registeredwith each other; aligns the images with a reference image; and adjuststhe alignment of the images with each other.
 15. The system of claim 14,wherein the software finds a transformation φ wherein:M _(total)(φ):=Σ_(j) M(U,V _(j)∘φ_(j))=max, subject toV _(total)(φ):=Σ_(i≠j) v(V _(i)∘φ_(i) ,V _(j)∘φ_(j))=min wherein:M(U,V_(j)∘φ_(j)) measures the similarity between the reference image Uand the transformed image V_(j)∘φ_(j); M_(total) is the sum of allM(U,V_(j)∘φ_(j)); v(V_(i)∘φ_(i),V_(j)∘φ_(j)) measures the variationbetween the two registered images V_(i)∘φ_(i) and V_(j)∘φ_(j) at theiroverlapped region; and V_(total) is the sum of allv(V_(i)∘φ_(i),V_(j)∘φ_(j)).
 16. The system of claim 15, wherein thesoftware finds the transformation φ by searching for a φ* such thatJ(φ*)=min using the equation:J(φ):=−M _(total)(φ)+λV _(total)(φ) wherein: λ is a constant to bedetermined.
 17. The method of claim 16, wherein the software finds φ* bycalculating the gradient ∇J(φ) and updating the search for the optimaltransformation φ according to:φ_(n+1)=φ_(n) −μ∇J(φ_(n)),μ>0,n=1,2,3, . . . wherein: μ is a constantused to control the convergence rate.
 18. The system of claim 15,wherein the software finds transformation φ (for φ(x)=x+u(x)) byminimizing:J(u ₁ , . . . ,u _(k))+rS(u ₁ , . . . ,u _(k))wherein:J(φ):=−M _(total)(φ)+λV _(total)(φ); wherein, λ is a constant to bedetermined; S is a regularization term; and r>0 is a regularizationparameter.
 19. The system of claim 18, wherein:S(u ₁ , . . . ,u _(k))=Σ_(j=1) ^(K)∫_(D) <B(u _(j)),B(u _(j))>dx,D⊂R ³wherein: B is a differential operator; and <,> denotes the inner productin L₂(R³); wherein L₂(R³) is the completion of the continuous functionswith respect to the L₂-norm.